 Submodularity of relative perimeter

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.